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Notes for: Starting from Randomness | Randomness in Class 3 Systems


*Additive [cellular automaton] rules

Of the 256 elementary cellular automata 8 are additive: {0, 60, 90, 102, 150, 170, 204, 240}. All of these are either trivial or essentially equivalent to rules 90 or 150.

Of all k^k^(2r+1) rules with k colors and range r it turns out that there are always exactly k^(2r+1) additive ones--each obtained by taking the cells in the neighborhood and adding them modulo k with weights between 0 and k-1. As discussed on page 955, any rule based on addition modulo k must yield a nested pattern, and it therefore follows that any rule that is additive must give a nested pattern, as in the examples below. (See also page 870.)


Note that each step in the evolution of any additive cellular automaton can be computed as

Mod[ListCorrelate[w, list, Ceiling[Length[w]/2]], k]

(See page 1087 for a discussion of partial additivity.)





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* Rule 218 [with simple initial conditions]
* Additive [cellular automaton] rules
* Generalized additivity [in cellular automata]
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From Stephen Wolfram: A New Kind of Science [citation] Previous note-----Next note